Download The distribution of the sample mean: an example Suppose that our

Probability and Statistics
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The distribution of the sample mean: an example
Suppose that our data consists of numbers
j = 1, . . . , 100000.
sin(j),
2
10000
4000
0
Frequency
The mean is μ ≈ 0.00 and the variance is σ ≈ 0.50.
Here is a histogram:
−1.0
−0.5
0.0
0.5
1.0
sin
Let us generate 1000 samples of size n = 100 (without replacement) from this data,
x1 , x2 , . . . , x100 ,
200
100
0
Frequency
and make a histogram for the observed values of the sample mean
x1 + . . . + x100
.
x=
100
−0.2
0.0
0.1
0.2
sample_mean
The average value of x-values is approximately 0.0024, which is close to μ, as it should be.
The variance of x-values is approximately 0.0052, which is close to σ 2 /100, as it should be.
Does the histogram appear normal to you? It is surely more normal than the underlying distribution.